A nonzero residual intersymbol interference (ISI) causes the symbol error rate (SER) to increase
where the achievable SER may not answer any more on the system’s requirements. Recently, a closed-form
approximated expression was derived by the same author for the residual ISI obtained by nonblind
adaptive equalizers for the single-input single-output (SISO) case. Up to now, there does not exist a
closed-form expression for the residual ISI obtained by nonblind adaptive equalizers for the single-input multiple-output (SIMO) case. Furthermore, there does not exist a closed-form expression for the SER valid for the SISO or SIMO case that takes into account the residual ISI obtained by nonblind
adaptive equalizers and is valid for fractional Gaussian noise (fGn) input where the Hurst exponent is
in the region of

We consider a nonblind deconvolution problem in which we observe the multiple output of a finite impulse-response (FIR) single-input multiple-output (SIMO) channel (unknown channel) from which we want to recover its input using adjustable linear filters (equalizers) and training symbols. In the field of communication, SIMO channels appear either when the signal is oversampled at the receiver or from the use of an array of antennas in the receiver [

In this paper we consider the nonblind adaptive equalizer where training sequences are needed to generate the error that is fed into the adaptive mechanism which updates the equalizer’s taps [

Recently [

In this paper, we propose a closed-form approximated expression for the residual ISI obtained by nonblind adaptive equalizers for the SIMO case (where SISO is a special case of SIMO). This new expression depends on the stepsize parameter, equalizer’s tap length, input signal statistics, SNR, channel power, and the number of receiving antennas and is valid for the fGn case where the Hurst exponent is in the region of

The paper is organized as follows. After having described the system under consideration in Section

The system under consideration, illustrated in Figure

The source sequence

The unknown subchannel

Each equalizer

The noise

Block diagram of a baseband SIMO communication system.

For

For

According to Figure

In this section, a closed-form approximated expression is derived for the residual ISI valid for the SIMO and fGn case. Based on this new expression, the SER is obtained.

In this subsection we derive the residual ISI valid for the SIMO and fGn case.

For the following assumptions:

the convolutional noise,

the source signal

the convolutional noise

the gain between the source and equalized output signal is equal to one;

the convolutional noise

the added noise is fGn as defined in the previous section in assumption

the Hurst exponent is in the range of

the residual ISI expressed in [dB] units may be defined as

The real part of

In this subsection, we derive the SER for a source signal

Based on the assumptions given in the previous subsection, the SER for the nonblind adaptive version valid for the SIMO and fGn case may be defined as

According to [

In this section we test our new proposed expression for the SER for the 16QAM case (a modulation using

Please note that channel cases A and B were also used in [

In the following we denote the SER performance according to (

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

SER comparison with the following parameters:

In this paper, we proposed for the real and two independent quadrature carrier case, closed-form approximated expressions for the residual ISI and SER obtained by nonblind adaptive equalizers that are applicable for the SIMO and fGn (

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to thank the reviewers for their helpful comments.